Path large deviations for interacting diffusions with local mean-field interactions

We consider a system of $N^{d}$ spins, with a local mean field type interaction. Each spin has a fixed spatial position on the torus $\mathbb{T}^{d}$ and a spin value in $\mathbb R$ that evolves according to a space dependent Langevin dynamic. The interaction between two spins depends on their spatial distance. We investigate the path large deviation principle from the hydrodynamic (or local mean field McKean-Vlasov) limit and characterise the rate function, for both the space dependent empirical process and the space dependent empirical measure of the paths. To this end, we generalize an approach of Dawson and G\partner. By the space dependency, this requires new ingredients compared to mean field type interactions. Moreover, we prove the large deviation principle by using a second approach. This requires a generalisation of Varadhan's lemma to nowhere continuous functions.